Mastering LRC (RLC) Circuit Responses

Problem statement and target quantities

Given a series RLC network with output measured across the capacitor, determine the transfer function H(s)=V_C(s)/V_{in}(s), classify damping from component values, and predict transient/frequency behavior.

Knowns: R, L, C, input model. Unknowns: poles, ω_0, ζ, Q, and response shape.

Assumptions and unit discipline

• Linear time-invariant component behavior around the operating point.
• Series topology with capacitor-output transfer definition V_C/V_{in}.
• Baseline Laplace derivation assumes zero initial current and capacitor voltage.
• Use SI units consistently: R in ohm, L in H, C in F, frequency in Hz or rad/s with explicit conversion.

Start With a Reproducible Baseline

Treat this worksheet like a lab handout: copy it, run it unchanged once, then change one parameter at a time. This makes cause-and-effect visible instead of mixing multiple design edits at once.

The baseline values are intentionally underdamped so you can clearly see how R controls ringing through zeta and Q.

Deriving the Transfer Function

We model a series RLC where $V_{in}$ is input and $V_C$ is output. The goal is not only to get $H(s)$, but to see exactly how each component affects the poles. Assume zero initial energy ($i(0)=0$) for the baseline derivation.

Method 1: Impedance Divider (Fastest)

In the Laplace domain, we treat components as impedances: $Z_R = R$, $Z_L = sL$, and $Z_C = \frac{1}{sC}$. The circuit becomes a simple voltage divider:

Substituting the impedances:

Multiply numerator and denominator by $sC$ to clear the fraction, yielding our standard form:

Method 2: KVL (First Principles)

Kirchhoff's Voltage Law states the sum of voltage drops equals the source:

$$ v_{in}(t) = v_R(t) + v_L(t) + v_C(t) $$

Substitute the constitutive relations $v_R = iR$ and $v_L = L\frac{di}{dt}$. Since $i = C\frac{dv_C}{dt}$, we can write everything in terms of $v_C$:

$$ v_{in}(t) = RC\frac{dv_C}{dt} + LC\frac{d^2v_C}{dt^2} + v_C(t) $$

Taking the Laplace transform gives the same $H(s)$ result as Method 1.

Mapping to Standard Form

In control form, second-order systems are written as:

To match our circuit equation to this, we divide our denominator $LCs^2 + RCs + 1$ by $LC$:

By coefficient matching, you extract the two parameters that matter most in practice:

1. Natural Frequency ($\omega_0$)

$$ \omega_0^2 = \frac{1}{LC} \implies \omega_0 = \frac{1}{\sqrt{LC}} $$

This sets the speed of the response.

2. Damping Ratio ($\zeta$)

$$ 2\zeta\omega_0 = \frac{R}{L} \implies \zeta = \frac{R}{2}\sqrt{\frac{C}{L}} $$

This determines the shape (oscillation). We can also express this using the Quality Factor $Q$:

$$ Q = \frac{1}{2\zeta} $$

The 3 Response Types

The roots of the characteristic equation determine behavior in both time and frequency domains. The sign of $\Delta = \zeta^2 - 1$ tells you whether energy exchange dominates (ringing) or dissipation dominates (non-oscillatory decay).

FAQ

Why do we care about poles?

Poles are the eigenvalues of the system. If they have an imaginary part (underdamped), energy sloshes between the Inductor (Magnetic field) and Capacitor (Electric field). If they are purely real, energy just dissipates in the Resistor.

How does increasing Resistance affect Zeta?

Looking at $\zeta = \frac{R}{2}\sqrt{\frac{C}{L}}$, increasing $R$ increases damping. Eventually, you cross 1.0 and the ringing stops completely.

Variable glossary and physical meaning

How to interpret the results

• zeta < 1 implies underdamped ringing; zeta ≈ 1 is near critical damping.
• Q quantifies resonance sharpness; higher Q means narrower, stronger peak behavior.
• Use the resistance-variant rows to see how damping shifts before changing component values on hardware.

Common mistakes and how to avoid them

• Using Hz formulas with variables defined in rad/s without conversion.
• Assuming ideal second-order behavior when ESR/ESL parasitics are significant.
• Ignoring initial conditions when validating transient simulations against measurements.

Worked example (interpreted)

When this model is invalid

How to validate your worksheet

• kvl_residual should simplify to zero.
• damping_metric=zeta-1 indicates under/critical/over damping sign.
• invalidity_proxy=f0/f_parasitic_ref rising means parasitic awareness should increase.

Domain-specific variants in this template

The resistance sweep rows (R_variant, zeta_variant, Q_variant) provide a fast design tradeoff path between ringing and settling time.

Practice problems (with answer outlines)

References and standards

• Oppenheim/Willsky, Signals and Systems (second-order system interpretation). MIT OCW companion.
• IEEE 1597 waveform-interchange context for simulation data workflows. IEEE reference.
• Horowitz & Hill, The Art of Electronics (practical damping/filter guidance). Reference site.